A008555 Primitive parts of Pell numbers.

1, 2, 5, 6, 29, 7, 169, 34, 197, 41, 5741, 33, 33461, 239, 1345, 1154, 1136689, 199, 6625109, 1121, 45697, 8119, 225058681, 1153, 45232349, 47321, 7761797, 38081, 44560482149, 961, 259717522849, 1331714, 52734529, 1607521, 1800193921, 39201

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also called Sylvester-Pell cyclotomic numbers. - Paul Barry, Apr 15 2005

According to Guy, Raphael Robinson noticed that a(7) and a(30) are squares and asked if there are more. There are no others in the first 10000 terms. [T. D. Noe, May 07 2009]

Examples

			a(8)=34 because pell(8)=408 and 408/(a(4)*a(2)*a(1)) = 408/12 = 34. [From _T. D. Noe_, May 07 2009]
		

References

• R. K. Guy, Unsolved Problems in Number Theory, A3.

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number. - _Paul Barry_, Apr 15 2005

Crossrefs

Cf. A061446 (primitive part of Fibonacci numbers). [T. D. Noe, May 07 2009]

Cf. A105606.

Programs

Mathematica

pell={1,2}; pp={1,2}; Do[s=2*pell[[ -1]]+pell[[ -2]]; AppendTo[pell,s]; AppendTo[pp, s/Times@@pp[[Most[Divisors[n]]]]], {n,3,40}]; pp (* T. D. Noe, May 07 2009 *)

Formula

a(n) = A000129(n) / Product_{dT. D. Noe, May 07 2009]

a(n) = Product_{k=1..n-1} if(gcd(n, k)=1, (1+sqrt(2))-(1-sqrt(2))*exp(2*Pi*I*k/n), 1), I=sqrt(-1). - Paul Barry, Apr 15 2005

Extensions

Corrected and extended by T. D. Noe, May 07 2009

Edited by N. J. A. Sloane, Oct 04 2009